The temperature range that a rocket can sustain depends on the design of the particular rocket.
Just to give you two extremes –
The shuttle's problem with cold temperature related to the O-ring seals on the solid rocket booster. That's a deadly problem when it gets as low as freezing.
Meanwhile the New Horizons spacecraft did a little trajectory correction between Pluto and Ultima Thule. It's VERY cold that far from the sun. No problem.
Different kind of propellant. Different rocket structure.
So, no problem running a rocket on the moon, if the rocket is designed and propellant chosen for that kind of environment.
Shuttle was not designed for the moon, or even for a freezing day in Florida.